Asset Co-movements: Features and Challenges

Nikolay Gospodinov*

Federal Reserve Bank of Atlanta

May 2017

Abstract

This paper documents and characterizes the time-varying structure of U.S. and international asset co-movements. It draws attention to the fact that while some of the time variation could be genuine, the sampling uncertainty and time series properties of the series can distort significantly the underlying signal dynamics. Proper transformation of the asset prices prior to extraction of common components is crucial for the validity and the robustness of the analysis. Frequency of the data also plays a role in the effectiveness of risk identification. We discuss some recent examples that illustrate the pitfalls from drawing conclusions from local trends of asset prices. On a more constructive side, we find that the U.S. main asset classes and the major international stock indices share a factor that is closely related to the business cycle. At even lower frequency, the common asset co-movement appears to be driven by demographic trends. Keywords: Cross-asset, within-asset and international asset co-movements; Rolling correlation; Time-variability; Persistence; Higher moments; Tail and asymmetric dependence; Risk factors; Sampling frequency.

JEL Classification : G13, G14, G17

* Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470; email: nikolay.gospodinov@atl.frb.org. This paper is prepared for the 22nd Annual Financial Markets Conference, “Managing Global Financial Risks: Shifting Sands and Shock Waves” (May 7–9, 2017). I would like to thank Richard Crump, Paula Tkac, and Larry Wall for helpful comments and suggestions. The views expressed here are the author’s and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System.

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1. Introduction

Identifying and quantifying the underlying co-movements in financial data have important

implications for asset allocation, portfolio valuation, regulatory enforcement, and policy analysis. Since

the 2008 financial crisis, there have been several episodes when the correlations between different

asset classes were elevated but dropped off quickly afterwards and exhibited substantial variability.

For example, the prevailing view in the last few years is of an increased correlation across asset classes;

that is, the “risk-on/risk-off” view of markets. Is this due to fundamental or transient (technical)

factors? Can we reliably isolate, in real time, the underlying source of risk from data and estimation

noise? Is the observed time-variability of these co-movements a true property of the data-generating

process or a symptom of misspecification that arises from omitted common factors and higher-

moment distributional features?

To answer these questions, it is desirable first to document and characterize the main regularities in

the asset co-movements across asset classes over time. While the scope of our empirical analysis is

limited to some frequently studied co-movements within and across asset classes, we still attempt to

provide some arguments about possible sources of common variation in asset returns, emphasize

some pitfalls in commonly used measures of these co-movements, and discuss the challenges of

measuring statistically the shifts of the distribution at various frequencies. The underlying sources of

common asset variations can be crudely classified into long-run, transient, and spurious factors. The

long-run (fundamental) sources of risk include structural macroeconomic and demographic factors at

business-cycle or even lower frequency. The transient factors are roughly interpreted as fluctuations

prompted by macro, political, and market-structure (technical) events that can force medium-term

asset reallocations and rotations and induce movements in the asset risk premia. Finally, the short-

term co-movements can arise from unanticipated (oil supply, for example) shocks or data noise. The

data noise and finite samples can generate possible spurious co-movements and observational

equivalence between the different factors, which renders a sharp distinction between the underlying

sources of risk almost infeasible. Despite these limitations, it is often tempting to ascribe

“fundamental” structure to some of the observed short-term movements as illustrated by some

examples discussed in this paper.

The discussion in the paper will be focused around several general observations. First, if there is robust

evidence that the asset co-movements are genuinely time-varying, then it would be useful to identify

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the source of this common variation. Market participants have suggested several sources of common

variation: regulatory changes and reduced market liquidity in some markets, correlated arbitrage,

growth of passive investment funds, particular investment strategies, and algorithmic trading, among

others. The heightened interdependence within and between asset classes can have important

implications for policymakers and can make the diversification and hedging implementation

increasingly challenging. On the one hand, the strengthened asset co-movements open up the

possibility of systematic market risk (such as concerns about the destabilizing effects of the “taper

tantrum” in 2013) and the proper calibration of monetary policy to economic conditions. On the other

hand, if monetary policy is a driver of the correlation change, then it is important to understand how

monetary policy decisions (and especially surprises) will affect markets; for example, how quantitative

easing could influence long-term bond and other asset prices. Identifying the sources of this increased

dependence in the joint distribution of asset returns may shed light on how to adapt to this shifting

and potentially unstable landscape.

Second, it should be acknowledged that another possibility for the increased time variation, and

potential instability, in the correlations across asset classes could be purely statistical due to the

limitations of the modeling framework and the use of second-moment measures.1 We argue in the

paper that particular attention should be paid to the sampling frequency and estimation uncertainty

associated with computing moments of the distribution, the effect of persistence on correlation-type

measures of co-movement, and differences in the correlation measures across different parts of the

distribution. While most drawbacks of the standard measures of co-movements are thoroughly

discussed and documented in the literature, they are often downplayed, due to their computational

and interpretational convenience, by practitioners. In this respect, it is surprising how

underappreciated the sampling and estimation uncertainty could be in forming investment and policy

decisions based on nonrobust measures of second-moment variation. As a result, there is a tendency

to over-analyze some high-frequency movements and attribute them to a particular underlying signal

without taking fully into account the data noise and estimation uncertainty. Unlike high-frequency co-

movements that are largely elusive and fairly transitory, low-frequency analysis provides a more robust

and reliable way to evaluate these dependencies and suggests that the asset common variation appears

1 There is a large and well-developed literature in financial econometrics on estimating dynamics conditional correlation models (see Engle, 2002, for example). To avoid technical details, we do not discuss this literature here. However, some of the remarks that we provide for time-varying correlations apply to this literature as well. For some caveats about dynamics conditional correlation models, see Caporin and McAleer (2013).

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to exhibit more stable relationships with macroeconomic and slowly moving demographic factors at

business-cycle or longer duration.

Third, the increased time variation in the correlations across asset classes could well be statistical and

arise purely from limitations of the modeling and conceptual framework and the use of second-

moment measures. There is now overwhelming evidence that higher moments of asset returns exhibit

features that cannot be reconciled with the dominating asset-pricing and portfolio-allocation models

in the academic literature, and the difference between the model-implied statistics and their empirical

counterparts is “anomalously” or “puzzlingly” large. For example, if sources of risk exhibit differential

impacts on the shape of the asset distribution over time (either by fattening the tail risk or shifting the

tail risk from one side of the distribution to the other), the information reflected in the second

moments will be incomplete and potentially misleading as it may exhibit substantial time variability

even when the underlying population correlations are constant. More broadly, if we seek to understand

how investors assess risk and reallocate their portfolios in response to the changing distribution of

asset returns and how their actions may themselves help to shape this distribution, more general

dependence measures—that reflect and summarize the information in the whole distribution—are

therefore necessary for identifying the underlying risk factors. The modern portfolio and asset-pricing

theory can then be modified accordingly to reflect this higher-order moment information.2

In light of these remarks, it is prudent to approach the analysis of asset co-movements by explicitly

acknowledging the model and estimation uncertainty surrounding all investment and asset-allocation

decisions. Since all financial models are constructed to approximate a complex reality, they are

inherently misspecified. This is often done intentionally, as parsimonious models draw only partial or

incomplete maps of the latent objects of interest either to emphasize particular aspects or because the

underlying structure is completely unknown. How large is the effect of this model misspecification on

the quantity of interest (for example, portfolio weights, stochastic discount factor) is an empirical

question. It appears that most of the anomalous and puzzling results in economics and finance tend

to diminish in importance when the prevalent analytical framework, based on linearity and multivariate

normality, is relaxed. Furthermore, while there is suggestive evidence of an increased asset-class

interdependence and a shifting market landscape, the exact source of this structural change (correlated

arbitrage, interdependencies, trading strategies, passive investment, algorithmic trading) is difficult to

2 Embrechts et al. (2002) provide a comprehensive discussion on the properties and pitfalls of correlation for measuring general dependence.

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pin down due to short samples, noisy data, and estimation uncertainty. Finally, since the object of

interest is often the joint distribution of all assets under consideration, more general and robust

measures of dependence, that embed information in the shape (asymmetry, tails) of this distribution,

are highly desirable, even though they suffer from similar drawbacks (noisiness and large estimation

uncertainty) as the correlation measure.

We approach this problem from an empirical perspective since economic theory provides only limited

guidance on the fundamental sources of time-varying co-movements across asset classes. There is a

well-established literature that studies the large directional swings in the covariance between U.S. bond

and stock returns in the post-World War II period. For instance, it is a stylized fact that this correlation

was largely positive between 1953 and the late 1990s but then turned negative, especially during the

2001 and 2007–09 recessions, with bonds providing a hedge against equity and macroeconomic risks.

Baele at al. (2010), Campbell et al. (2015), David and Veronesi (2016), and Song (2017), among others,

propose models that better fit the observed dynamics of the correlation between bond and stock

returns. However, developing a unifying framework for modeling jointly several major asset classes

(stocks, government bonds, corporate bonds, commodities, currencies) proves to be prohibitively

difficult. Cochrane (2015) identified the question “Once we find the factor structure in bonds, what

is the factor structure of expected returns across asset classes?” as one of the “bigger” questions facing

financial economics. On that note, dimension reduction techniques such as factor analysis prove to

be useful tools for summarizing the common variation across asset markets. Identifying the main

sources of risk via the estimated factors is of fundamental importance not only for guiding investment

decisions but also for monitoring financial stability, stress testing, and more.

While distinguishing among fundamental, transitory, and spurious sources of risk proves empirically

difficult, we find clear evidence of long-run co-movements at business-cycle or even lower frequency.

There is a reasonable basis for tying these co-movements to macroeconomic fundamentals and

demographic factors. While there is also some evidence of co-movement at the shorter end, one

needs to be careful in analyzing this evidence as there are several sources of possible error that make

the competing hypotheses observationally equivalent. These short-term co-movements, which force

asset reallocations and induce temporary co-movements in the asset-risk premia, tend to be more

unstable. Nevertheless, there is a tendency to ascribe more fundamental structure to these variations,

which warns against overreliance on various nonrobust measures of co-movement that are routinely

used for portfolio allocation, evaluation of asset–pricing models, and factor investing.

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The rest of the paper is organized as follows. Section 2 provides a motivating example that exposes

the potential challenges and biases when analyzing co-movements based on recent local-trend, high-

frequency data. Section 3 discusses a few statistical problems that could affect the reliability of rolling

second-moment measures of dependence. In particular, we discuss the effects of the persistence of

the individual series, the estimation and sampling uncertainty, and the sensitivity of these measures to

changes in the higher moments of the distribution. Section 4 attempts to extract the common price

variation across asset classes and relates them to business cycle fluctuations and longer-run

demographic trends. This allows us to draw some tentative conclusions about these co-movements

based on future projections of these business and demographic cycles. Section 5 offers some

concluding remarks.

2. An Illustrative Example: Oil Price Co-movements

From the beginning of 2014 to the middle of 2016, the price of crude oil dropped by more than 50

percent, a drop that was accompanied by a substantially elevated correlation between oil price and

several other asset classes. This unusually high correlation attracted the attention of market analysts

for at least two main reasons. First, this co-movement significantly restricted the set of diversification

and hedging opportunities in the market. Second, oil prices as the driving factor behind the asset co-

movements on a more sustained basis may pose risks to the stability of the financial system given the

highly volatile nature of commodity prices, with their larger exposure to geopolitical risk, supply

disruptions, and more.3

Figure 1 plots the dynamics of several asset prices—5-year, 5-year forward breakeven inflation from

Treasury Inflation-Protected Securities (TIPS) and Treasury prices, U.S. dollar index (DAX), and one-

year changes in the S&P 500 index, and Barclays high yields index—versus the log oil price from the

beginning of 2014 to the end of February 2017. Ending in May 2016, the series (or some

transformations of them) are plotted as solid lines to illustrate the seemingly tight relationship between

the oil price and other asset prices that was widely documented and discussed by pundits and the

media at that time. In contrast, the dashed lines after May 2016 tend to suggest that this relationship

has substantially weakened.

3 The median values of the option-implied volatility indices OVX and VIX (for oil prices and S&P 500, respectively) since the beginning of 2014 (2015) are 37.78 (42.28) and 14.12 (14.48).

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Figure 1. Oil price dynamics against (1) 5-year, 5-year forward breakeven inflation; (2) U.S. dollar index (DXY); (3) percent changes in S&P 500 index from a year ago; and (4) Barclays high yield index. The solid lines denote the dynamics of each series from January 2014 to May 2016, and the dashed lines represent the period June 2016 to February 2017.

Was this co-movement due to fundamental common factor/information driving all these variables

(global demand shock, for example), or was it caused by transitory (market-specific) factors reflected

only in the asset risk premia? While seeking answers to these questions, we want to emphasize some

statistical features—such as the possibility of spurious time variability due to persistence (trending

behavior) of the variables over this particular period or sensitivity to movements in other parts

(skewness, tails) of the joint or marginal distributions—that are often overlooked by practitioners.

To gain a better understanding of these issues, we focus on the relationship between breakeven

inflation (BEI), embedded in nominal and inflation-protected securities, and oil prices from the

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beginning of 2014 to the middle of 2016. During this particular period, both oil prices and BEI

exhibited a downward trend, and this strong correlation was largely attributed to the effect of oil prices

on inflation expectations. This interpretation was challenged on several grounds (see, for example,

Gospodinov, Tkac, and Wei, 2016). First, BEI is not a clean measure of inflation expectations as it

also contains other, unobservable components such as liquidity and risk premium. Second, it is

somewhat counterintuitive for the day-to-day (transitory) variations in oil prices to affect inflation

expectations 5 to 10 years out. Finally, both variables are highly persistent and the sample correlation

may provide a spurious signal of a relationship between the two variables.

A proper decomposition of the observable BEI into its latent components (see Gospodinov and Wei,

2015) reveals the following. Despite the wide variations in BEI, long-term inflation expectations

appear to be stable and uncorrelated with oil prices. A large portion of the low-frequency variation in

BEI can be attributed to the inflation risk premium. But most of the high-frequency variation in the

recent dynamics of BEI can be attributed to a “liquidity” premium. This “liquidity” factor captures a

wide range of market-related factors such as seasonal carry, deflation floor, limits to arbitrage, tenor-

specific liquidity, redemptions, reallocations, and hedging in the TIPS market following an oil price

drop or global financial turbulence. For example, an event that drives flight-to-quality into nominal

Treasuries and/or forced sales of TIPS leads to a lower BEI without any change in inflation

expectations.

Figure 2 shows that almost all of the recent correlation between BEI and oil prices is being picked up

by the “liquidity” premium. As shown above, other asset classes (stock prices, high-yield bonds,

municipal bonds, exchange rate) have also exhibited an elevated correlation with oil prices during this

period. If genuine, it is useful to determine if this is due to a global demand driver, correlated risk, or

market specificities (such as forced de-risking and liquidations, reallocations, flight-to-safety, or

covering hedges).

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Figure 2. Time plot of weekly TIPS liquidity premium with oil prices and oil option-implied (OVX) volatility

For example, numerous studies have documented the existence of persistent mispricing in various

asset markets. It is conjectured that common factors could drive this mispricing and the resulting

arbitrage. Brunnermeier and Pedersen (2009) argue that availability of funding may have liquidity

effects on asset prices. Also, if capital returns slowly to the fixed-income funds following a period of

flat performance, then the arbitrage in various fixed-income markets (corporate, CDS, Treasury, TIPS)

can exhibit commonalities (Fleckenstein, Longstaff, and Lustig, 2014). Finally, the macroeconomic

environment after the financial crisis may also have contributed to the strengthened asset co-

movements. For instance, Datta et al. (2017) provide evidence that oil and equity returns have become

more responsive to macroeconomic news during the zero lower bound period.4

While this and other (more direct or anecdotal) evidence on the 2014–16 episode suggests that the co-

movement of various asset prices and oil price can be attributed to market-structure factors, the next

section attempts to raise the awareness that a part of this co-movement can be purely coincidental or

unreliable. In particular, we explore the role of persistence of the series under considerations,

estimation, and sampling uncertainty, as well as the potential limitations of second-moment based

measures of co-movement to represent accurately changes in the joint distribution of interest.

4 There is also evidence that the transmission mechanism of propagating the oil shocks through the US economic system has changed due to the increased role of domestic oil production (Baumeister and Kilian, 2016).

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3. Statistical Challenges for Measuring Asset Co-movements

3.1 Persistence and Measures of Co-movement

As illustrated above, the co-movement across asset classes is often analyzed using (commodity, for

example) prices and yields. While commodity prices and bond yields are usually modeled as mean-

reverting processes, they are strongly persistent with a very slow mean reversion. Because the high

persistence generates local trends, these local trends can induce spurious correlation even when the

underlying processes are completely unrelated to each other. Furthermore, even when the processes

are genuinely related, the serial correlation present in prices or yields can obfuscate the underlying

relationship if the serial correlation is not properly taken into account.

To illustrate the latter point, we examine the leverage effect (the negative relationship between stock

prices and volatility) that occupies a prominent place in financial economics and econometrics. Figure

3 plots the time series dynamics of the S&P 500 (SPX) and VIX indices (implied by options written

on SPX) from January 2000 to February 2017.

Figure 3. Time series dynamics of SPX and VIX indices

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While the negative relationship is evident during several episodes of sharp stock declines, there are

periods (the beginning of the sample as well as the 2012–16 period) when any possible co-movement

may be masked by the trending behavior of the SPX index.

Figure 4 below suggests that the 120-day sample correlation between the SPX and VIX (blue line) is

highly volatile with even positive values over some sub-samples. It is tempting to conjecture that

removing the local trends in the SPX index by transforming it to SPX log returns may stabilize and

uncover better the leverage effect. The red line in figure 4 reveals that this transformation reduces the

variability of the sample correlation coefficient, but it leads to a severely biased measure that

underestimates significantly the negative co-movement between the SPX and VIX. This is due to the

fact that the SPX returns and the VIX index are highly unbalanced in terms of their statistical

properties, since the former is an uncorrelated (or only weakly correlated) process and the latter is a

highly persistent and bounded variable.

Figure 4. 120-day correlation between SPX and VIX

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To render the statistical properties of both series similar, we compute the log changes of the VIX

index. Figure 4 plots the 120-day rolling correlation5 of SPX returns and VIX changes in yellow. In

fact, this is akin to the way the leverage effect is modeled in financial econometrics. The correlation

now is much more stable and averages around -0.8, which is the magnitude of the leverage effect that

is typically estimated in the empirical literature. Note that some of the variability in the rolling

correlation coefficient can be attributed to overlapping estimation uncertainty as discussed below.

To illustrate the point that the observed correlation may be spurious, we generate data from two

uncorrelated, near-unit root processes (that match the persistence in bond yields and oil prices) and

plot in figure 5 one realized sample path of the two processes. The unconditional correlation between

the two series is 0.36 with an even stronger correlation over sub-samples that arises from “common”

local trending behavior. Any observed commonality is completely spurious, as the two series are

generated as independent of each other.

Figure 5. Time plot of two independent highly persistent processes

5 In this paper, we use Spearman’s (rank) correlation instead of the standard Pearson’s correlation coefficient. This choice is dictated by the robustness properties of the rank correlation. Also, when appropriate, we computed the correlation coefficient using standardized returns using a GARCH(1,1) model for estimating the conditional variance of returns. The results were very similar and are not reported, unless specified otherwise.

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This should serve as a warning sign in analyzing and justifying observed co-movements of highly

persistent processes. It also suggests that one should avoid computing correlations in levels and

analyze instead the transformed series after the persistence is removed. This is the approach that we

follow in the rest of the paper.

3.2 Time Variability and Estimation Uncertainty

The last decade has witnessed a proliferation of passive investment funds. Since 2005, the share of

S&P 500 ownership of passive mutual funds and exchange-traded funds (ETF) increased from 4.6

percent to 11.6 percent (Wall Street Journal, October 2016) while the share of active mutual funds and

ETFs remained flat around 17 percent. This same period has been characterized by an increased

correlation among U.S. equities due to convergence of equity betas of individual stocks. Passive

investing is believed to have benefited and possibly contributed to this empirical regularity, although

this relationship is confounded by other factors that include proliferation of electronic and algorithmic

trading, globalization, unconventional monetary policy, and reduced market liquidity arising from

regulatory changes in the developed countries. This type of observation is typically made based on

computing some time-varying correlation coefficient and following its dynamics over time.

Figure 6 presents the 60-day and 120-day rolling correlations between large-cap (S&P 100) and small-

cap (S&P 600) returns from January 2000 to February 2017. The figure reveals substantial variability

in these correlation coefficients with occasional sharp upward and downward turns (for example, the

one that occurred in the beginning of this year) that could affect directly the inflow or outflow of

funds and performance of passive and active investment strategies. What is less discussed, however,

is the substantial estimation uncertainty surrounding these estimates. There are two main sources of

this uncertainty. First, it is the length of the sample and the sampling frequency, as short samples and

high frequency data may be preferred to ensure that the recent momentum in the series is followed

nearly in real time. Second, rolling correlations involve overlapping observations and a large estimation

error can be accumulated and amplified and, combined with the short sample size, can persist.6

6 Rolling estimators can also be interpreted as nonparametric estimators (Ang and Kristensen, 2012; Adrian et al., 2015) with a slow rate of convergence and larger variability.

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Figure 6. Rolling correlations of S&P 100 and S&P 600 (small cap) returns

It is instructive to investigate further how much of the time variation of the rolling correlation

coefficient can be attributed to estimation and sampling uncertainty. For this purpose, we generate

artificial data that is calibrated to the actual data by matching the sample size, estimated means,

unconditional covariance, and conditional variances of the S&P 100 and S&P 600 returns. The

standardized returns are drawn from a bivariate t-distribution with varying tail thickness (a degree of

freedom parameter that is drawn from a uniform distribution on the interval [4, 13]) but a constant

correlation parameter. Figure 7 plots the 60-day and 120-day rolling correlations for the simulated

data. Interestingly, these correlations exhibit similar time variation as the correlations in figure 6 even

though the true correlation is constant. Part of this variation is due to changes in the tails of the

distribution, arising from the time-specific degrees of freedom parameter of the t-distribution, but

most of the variation is a result of overlapping estimation uncertainty. While this is only suggestive, it

warns against overreliance (without properly taking into account the estimation uncertainty) on rolling

estimation that is routinely used for portfolio allocation, evaluation of asset-pricing models, and factor

investing (for a recent example, see Asness et al., 2017).

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Figure 7. Rolling correlations for simulated returns data

DeMiguel et al. (2009) have demonstrated that the estimation imprecision may adversely affect asset

allocation decisions. We mimic their argument by computing out-of-sample Sharpe ratios from an

optimal mean-variance portfolio with estimated weights7 and the naïve portfolio with equal (1/N,

where N is the number of assets) weights. The asset returns, which are also used later in this paper,

are for four major U.S. asset classes—– S&P 500 index (SPX) returns, Bloomberg Barclays Treasury

total returns, Goldman Sachs commodity index (GSCI) returns, and USD index (DXY) returns—as

well as five international equity indices (converted in USD)—S&P 500 (SPX), FTSE 100 (UKX),

Nikkei 225 (NKY), DAX, and MSCI emergent markets (MXEF) index. The data are daily from

January 2000 to February 2017. We use a rolling window sample of 500 daily observations for

estimation of the weights for the mean-variance problem. These estimated weights, as well as the fixed

7 The recent literature has suggested various ways to sharpen the estimation of the portfolio weights under the mean-

variance risk measure. Another way is to replace the expectation bounded risk (such as the mean-variance risk measure) with a more robust, which takes explicitly into account the tails of the distribution, coherent risk measure such as the conditional value-at-risk (Assa and Gospodinov, 2014). Coherent risk measures are tightly linked to the Choquet expected utility, which can distort the probability of different events by assigning, for example, larger weights to less favorable events and smaller weights to more favorable ones (see Bassett et al., 2004).

15

1/N weights, are then interacted with the next-day return and these out-of-sample returns are used to

compute the corresponding Sharpe ratio. The Sharpe ratios for the U.S. across-asset portfolio are

0.205 for the equal-weight and 0.160 for the mean-variance portfolio, with the difference being

statistically significant at the 5 percent significance level (p-value of 0.028). For the international

portfolio, the Sharpe ratios are 0.274 for the U.S. across-asset portfolio and 0.261 for the international

portfolio, with the difference between the Sharpe ratios being statistically insignificant. This is in line

with the results in DeMiguel et al. (2009) showing the effects of estimation error on optimal portfolio

allocation.8

Summarizing the time variation of the correlation structure of more than two asset returns, which is

the case for the two portfolios considered above, is not as trivial as the bivariate correlation coefficient.

To do this, we construct a distance that measures similarities between two correlation matrices over

different time periods. Let P denote the correlation matrix of asset returns for the period t to Pt

, and R denote the correlation matrix of asset returns for the period Pt to RPt . Then, we

compute the correlation matrix distance (Herdin et al., 2005) as

where )(tr is the trace operator and denotes the Frobenius norm. This distance metric takes values

between zero (when the two correlation matrices are equal) and one. In what follows, we set 240P

and 120R .

The left panel of figure 8 plots the time variability of the correlation matrix for the four U.S. asset

returns. While there was a change in the correlation matrix during the 2007–09 recession, the largest

jump occurred in 2013. For the sake of comparison, the right panel of figure 8 presents the correlation

matrix distance computed from 2-, 5-, 10- and 30-year bond yield changes (intentionally plotted on

the same scale as the right graph). Due to the tight restrictions imposed by the term structure of

interest rates, the bond correlation matrix exhibits very little time variation.

8 It may be useful to remind the readers that any asset return can be decomposed (by identity) as the product of its sign and its absolute value (Anatolyev and Gospodinov, 2010, 2015). If the commonality is primarily in the sign (directional) changes across assets but not in the volatility, then this could be another reason why the fixed-weight portfolio may dominate the mean-variance optimization.

,)(

1),(RP

RPRP

trd

16

Figure 8. Time-varying changes in the correlation matrix distance measure of U.S. asset classes (left) and bond returns (right)

In summary, one should exercise caution when the analysis is based on rolling correlations with a

relatively short sample window, as seemingly pronounced time variability can be due to estimation

and data noise.

3.3 Robustness and Sensitivity to Time-Variability in Higher Moments

It is well known that the unconditional marginal and joint distributions of financial asset returns have

pronounced non-Gaussian features. As we argued in the previous section, part of the observed time

variability in the second-moment statistics (covariance, correlation) can be attributed to movements

in the higher moments of the distribution if the latter are ignored and not adequately modeled. Later

in this section, we discuss a more holistic (information-theoretic) approach to studying dependence

across assets that incorporates information in the whole distribution.

Before we do this, we adopt a simpler, more descriptive method for characterizing the dependence in

different parts of the distribution. It has been documented (Ang and Chen, 2002; Longin and Solnik,

2001; Karolyi and Stulz, 1996; among others) that the correlations increase, often dramatically, during

extreme downward market movements. These increased correlations diminish the benefits of

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portfolio diversification and hedging, especially in situations when they are needed the most.9 More

generally, we would like to compute robust dependence measures along the quantiles of the

distribution, including the tails. Let txr , and tyr , ( Tt ,...,1 ) denote the ranks of the pair of

(standardized) asset returns x and y . Using this notation, Spearman’s rank correlation is defined as

),( ,, tytx rrCorr . The sample quantile dependence at quantile is constructed as (Patton, 2013)

,15.0 ,},{

,5.00 ,},{

,,1)1(1

,,11

qqrqrI

qqrqrI

tytx

TtT

tytx

TtT

where I is the indicator function. This definition can be easily modified to quantile or exceedance

correlations based on the Spearman’s rank correlation.

Figure 9 plots the quantile dependence between four international stock returns (FTSE, NKY, DAX

and MSCI emerging markets) and the S&P 500. The computation is based on standardized daily

returns for the period January 2000–February 2017. For these stock returns, there is no pronounced

dependence asymmetry in the tails of the distribution. The dependence is largest in the middle of the

distribution and drops off in the tails. If the interest lies in a particular quantile of the distribution, the

quantile dependence can be computed over a rolling window and supplement the information for the

time-varying pair-wise correlation coefficients. The quantile (exceedance) correlation is characterized

by similar level and pattern.

9 For some drawbacks and biases in evaluating and testing the change in the correlation during market turmoil, which is accompanied with elevated asset volatility, see Boyer et al. (1999), Campbell et al. (2008), and Forbes and Rigobon (2002).

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Figure 9. Quantile dependence between international stock returns (FTSE, NKY, DAX and MSCI emerging markets (MFEX)) and the S&P500 for the period January 2000 – February 2017

Computing general dependence measures, instead of correlation coefficients, could help us to gain

some understanding of the level of dependence and the stability of the relationship between the

different asset classes. This has implications for the approach to asset allocation and evaluation of

systemic risk. In what follows, we focus on pair-wise or bivariate dependence and co-movement. One

approach to studying the dependence between two random variables x and y is to resort to Sklar’s

theorem, which states that there exists a unique copula C with dependence parameter such that

the joint distribution ),( yxF of the two variables can be expressed as

),),(),((),( yHxGCyxF

where )(xG and )(yH are continuous marginal distributions. When )),(),(( 11

2 yxC

,

where and 2 denote the marginal and bivariate Gaussian cumulative distribution functions, then

becomes the standard correlation coefficient. Other choices of a copula function are also possible.

A more general approach to modeling dependence is based on the generalized contrast or maximum

19

entropy principle. Let P and Q be two probability measures; for example, two probability measures

associated with two asset returns or the physical and risk-neutral probability measures. One way to

measure the divergence between the two measures (Csiszár, 1972) is to solve the following

optimization problem

,),( dQdQ

dPQPD

where ),0[: R is a convex, continuously differentiable function such that 0)1()1( .

The measure D is nonnegative and 0),( QPD if and only if the two measures coincide, QP .

The function is often assumed to belong to the Cressie-Read (1984) divergence family

11

)1(

1)(

1

aa

a .

One celebrated member of this divergence family is the Kullback-Leibler divergence, which is

obtained by setting 0 and takes the form

.0for 1)ln()( aaaaa

To illustrate the usefulness of this approach, suppose that e

iR denotes the excess return on the risky

asset i ( Ni ,...,1 ), P signifies the data-generating measure, and Q is the risk-neutral measure. The

risk-neutral measure Q with minimal entropy relative to the physical measure P can be obtained as

the solution to the problem (Stutzer, 1995)

dP

dQQ Q

Q

ln argmin E ,

subject to the no-arbitrage restriction

.,...,1 ,0][ NidQRR e

i

e

i

Q E

The solution Q gives rise to the following density

,)][exp(

)exp(

1

1

e

ii

Ni

e

ii

Ni

Rw

Rw

dP

dQ

E

where the density parameters (portfolio weights) ),...,( 1

Nwww are the solution to the problem

)].[exp(lnargmin 1),...,( 1

e

ii

Ni

www

RwwN

E

20

One interesting observation is that )][exp(ln 1e

ii

Ni Rw E is the cumulant-generating function of

e

ii

Ni Rw 1 , which characterizes all the information in the distribution of the excess returns. When

excess returns are assumed to be multivariate normal, all cumulants beyond the first two cumulants

are zero and the above optimization problem collapses to the usual mean-variance portfolio problem

wRCovwwRw ee

www N

][5.0][argmin),...,( 1

E ,

which admits the closed-form solution ][][ 1 ee RRCovw E . Substituting for w , the relative

entropy bound collapses to the scaled Hansen-Jagannathan (1991) bound ][][][5.0 1 eee RRCovR EE

which in the case of one asset becomes the squared Sharpe ratio of this asset return. Some numerical

calculations suggest that a large part of the entropy is accounted for by the higher-than-second

cumulants that arise from the non-Gaussianity of the excess return data. Thus, ignoring these higher

moments in measuring entropy and dependence will result in a significant misspecification and

spurious dynamics in the first two moments. Once higher moments and more general loss/risk

functions are allowed for, most “anomalies” and “puzzles” tend to diminish in terms of magnitude

and economic significance.

A more convenient measure of general dependence between two asset returns is based on the

Hellinger distance measure obtained (up to a scale) from the Cressie-Read family of divergence

measures above by setting 2/1 .10 Let ),( yxf be the joint density and )()( yhxg be the

product of the marginal densities of the asset returns x and y . In this case, the Hellinger dependence

measure is given by

.)()(),(2

1 2

dxdyyhxgyxfDh

If x and y are independent, )()(),( yhxgyxf and 0hD ; otherwise 0hD . Conveniently,

when x and y are bivariate normally distributed, 0hD if 0 and 1hD if 1|| , where

again denotes the correlation coefficient. Finally, the Hellinger distance is directly related to the

copula (Granger, Maasoumi, and Racine, 2004)

,)],(1[ 2/1 dudvvucDh

10 Unlike the other measures in the Cressie-Read divergence family, the Hellinger distance is a proper measure of distance since it is positive and symmetric, and it satisfies the triangular inequality.

21

where vuvuCvuc /),(),( 22/1 and )(C is the copula function introduced above.

This dependency measure can be easily accommodated to account for time variation and asymmetric

dependence (Jiang, Wu, and Zhou, 2016). While we do not provide empirical results based on the

Hellinger measure in this paper, we recommend that the statistics discussed in this section be used to

augment the standard procedures based on second-moment information. We refer the interested

reader to Gospodinov and Maasoumi (2017) for an application of this general approach to aggregating

information from potentially misspecified asset-pricing models.

4. Risk Factor Extraction

It is widely believed that financial asset returns contain a small low-frequency, persistent component;

see, for example, Bansal and Yaron (2004) for the role of long-run risks in explaining the equity

premium puzzle. Despite its theoretical appeal, the empirical evidence on the existence of such long-

run components is rather weak, as the observed asset returns do not appear to exhibit any persistence.

The potential explanation for this tension between theory and empirics is that the underlying slow-

moving component is overwhelmed by higher frequency noise and volatility, and quantifying its

impact and empirical detection prove challenging using only time series data on intrinsically more

volatile asset returns. However, provided that this trend component is common across assets, one

could use a cross-section of asset returns and estimate their common variation by the method of

principal components. Thus, cross-sectional information can be useful to extract more precise signals

about common variation and factors.

Unlike the correlation and dependence measures discussed in Section 3 that apply largely to pair-wise

relationships and can exhibit some idiosyncrasies, principal components and factor analysis are a

convenient tool to summarize the co-movements in a large cross-section of asset returns. Another

advantage of this approach is that the precision of the estimation increases with the dimension of the

asset returns included in the analysis. Below, we attempt to isolate common low-frequency movements

in asset returns and relate them to some macroeconomic and demographic factors.

4.1 Business Cycle Co-movements

Here, we follow Bai and Ng (2004) by estimating the common factor from returns and then integrate

22

the process to obtain the common stochastic trend component.11 This method guards against the

possibility of spurious trends and is particularly effective when the number of assets is large. An

alternative approach is to model the common variation in asset prices through cointegrating vectors.

Given the differences in the persistence of the individual series in levels and the lack of robustness of

the co-integration approach to deviations from the exact unit root for each process (Elliott, 1998), we

have adopted the principal component estimation of Bai and Ng (2004).

We first analyze the factor structure in the four U.S. asset classes considered in Section 3: S&P 500

index (SPX), Bloomberg Barclays Treasury total return index, Goldman Sachs commodity index

(GSCI), and USD index (DXY). All series are daily returns for the period January 2000–February

2017. The common factor is estimated from these returns by the method of principal components,

and figure 10 plots the integrated and linearly detrended process.

Figure 10. Left: Common variation across asset classes with shaded areas representing NBER-dated recessions. Right: R2 from projecting individual asset returns on the common factor

The estimated factor exhibits sharp decreases during recessions, with the drop during the 2007–09

recession being particularly pronounced. The factor also appears to exhibit some higher-frequency

cycles, but they are more difficult to identify. The right panel in figure 10 reports the R2 from projecting

the individual asset returns on the common factor. The largest loading on the factor is for the S&P

500 index, although bonds and commodity prices also contribute significantly to the common factor

variation.

11 The integrated process is linearly detrended and demeaned.

23

It would be interesting to see if this common variation is specific to the United States or if it reflects

some global factor as well. For this purpose, we compute the common factor from international stock

returns on S&P 500, FTSE in UK, Nikkei in Japan (NKY), DAX in Germany, and MSCI emerging

markets index (MXEF) in both USD and local currency. Figure 11 plots the common factors.

Interestingly, these returns share very similar dynamics with the U.S. common factor. Moreover, the

largest contributors to these dynamics are FTSE, DAX, and MXEF and not S&P 500. Consistent with

figure 9, Nikkei appears to have a large idiosyncratic component. Also in line with figure 10 where

DXY exhibits little correlation with the common factor, the common variation in USD and local

currency is fairly similar. The results in figures 9 and 11 suggest a substantial integration of the

international equity markets in the post-2000 period that is accompanied with a decline in the benefits

from global diversification (see also Cotter et al., 2016).12

Figure 11. Left: Common variation in international stock returns with shaded areas representing NBER-dated recessions. Right: R2 from projecting individual stock returns on the common factor

Despite the fairly short time span of the data, it would be instructive to relate more convincingly this

common variation with some underlying macroeconomic signal or risk factor. To provide some

suggestive evidence, figure 12 presents the Atlanta Fed/New York Fed turning point indicator of the

labor market,13 along with the Chicago Fed national activity index (CFNAI). The labor market

indicator is computed from vintage data and is available “almost” in real time. It is particularly useful

12 Pukthuanthong and Roll (2009) provide an insightful analysis on the flaws of correlation as a measure of global market integration. Instead, they recommend the use of global factor exposure for gauging financial market integration.

13 This is based on ongoing joint work with Richard Crump and Ayşegül Şahin from the Federal Reserve Bank of New York.

24

for monitoring since it is a one-sided filter with no distortionary effects on the cyclical dynamics. The

CFNAI series is relatively noisy and is constructed with data that are released with a delay. While the

smoothing in the Atlanta Fed/New York Fed indicator induces a phase shift in the series, it provides

a much improved estimate of the turning points and the local trends in the underlying signal.

Figure 12. Chicago Fed national activity index (CFNAI) and AtlFed/NYFed labor market indicator. Shaded areas are NBER-dated recessions

There are a couple of observations that warrant some remarks. First, the smoothed labor market

indicator identifies the turning points of the business cycle in advance, dating the recessions and the

expansions, which would be of great value to policymakers. The noisy monthly reports in the data-

dependent policy or the volatile asset dynamics in asset allocation can be supplemented and validated

with this lower-frequency, local-trend information. Moreover, it is striking how closely the smoothed

labor indicator underlies the dynamics of the common variation in asset returns, suggesting the

presence of a strong business cycle component in both domestic and global financial asset prices.

Pinning down these business cycle components in asset prices has some important implications for

medium-term investment and policy decisions. For example, this information can be used in preparing

25

supervisory scenarios for annual stress tests by the Federal Reserve. More specifically, a factor-

augmented vector autoregressive model, with an asset-pricing factor estimated as above, can be used

to generate conditional forecast paths (see Waggoner and Zha, 1999), under various scenarios, for the

variables of interest. Incorporating this asset-pricing factor, using both domestic and international

data, should further improve the fit of the model and the accuracy of the conditional forecasts.

4.2 Long Cycles: Low Frequency Information in Demographic Variables

In this section, we investigate if there is any other common variation in asset prices—and stock prices

in particular—at even lower frequency than that of the business cycle. Recent literature has established

the usefulness of low-frequency demographic variables for long-horizon stock market returns. This is

because savings rates, and possibly risk preferences, vary substantially over the life cycle, with savings

rates peaking in middle age and then being drawn down in old age. These savings directly impact the

pool of funds available for investment in the stock market. In fact, they may explain and predict some

of the very persistent, low-frequency movements in stock market valuation ratios, such as the dividend

or earnings price ratios.

To try to address this issue, we use annual data for the period 1946–2015 for the S&P 500 returns,

returns on long-term and intermediate-term U.S. government bonds, changes in S&P 500 price-

dividend ratio, changes in S&P 500 dividend yield, and changes in S&P 500 earnings-price ratio.14 The

data for the government bond returns are from Ibbotson Associates and the stock price data are from

Robert Shiller’s and Amit Goyal’s websites.

We also collect annual demographic data from the Census Bureau and construct the variable middle-

young (MY) ratio as the ratio of middle-aged (40–49) and young (20–29) cohorts. In addition to the

historical data, the Census Bureau provides projections until 2060. Figure 13 plots the smoothed

common factor estimated from the asset returns, as described in the previous section, along with the

middle-young ratio.

14 More specifically, dividends and earnings are 12-month moving sums. The dividend-price and earnings-price ratios are

constructed as the difference between the log of dividends (earnings) and log of prices. The dividend yield is the difference between the log of dividends and log of lagged prices. Due to the near-unit root behavior of these series, we work with their first differences.

26

Figure 13. Common factor in asset returns and middle-young ratio

During the 1946–2015 period, the long-cycle factor in asset returns seems to co-move closely with the

demographic trend, with a common trough in the early 1980s and a common peak in the early 2000s.

This is consistent with other results in the literature on the relationship of the demographic variables

with stock valuation ratios (Favero et al., 2011) and interest rates (Favero et al., 2016).

A unique feature of the demographic variables is that their low frequency trends can be projected with

reasonable accuracy decades into the future. The projections of the MY ratio by the Census Bureau

until 2040 (represented on the graph with a dashed line) thus facilitate the forecasting of both the

market valuations and the small low-frequency component of market returns. The MY ratio is

projected to fall until 2020, which implies a downward pressure on the stock valuation ratios and

interest rates due to demographic factors. After 2020, the MY ratio starts to increase again until 2040,

when it reaches a new peak. It is expected that during this period, the downward pressure on valuation

ratios and interest rates from demographics is diminished and even reversed.

27

It is instructive to discuss further the implications of the demographic-trend projections on long-

horizon forecasting for stock returns and valuation ratios.15 The literature has explored many

predictors for asset returns, with varying and often debated success (for a comprehensive review, see

Goyal and Welch, 2008). However, the future trajectory of most predictors is itself highly uncertain,

making them less useful for longer-term forecasts. Also, the innovations from the predictive regression

for stock returns and the dynamic model for the predictor are often strongly correlated. For example,

when the lagged dividend-price (dp) ratio is used as a predictor, this correlation is -0.95 implying that

a fall (rise) in the dp ratio is associated with positive (negative) returns. But this strong

contemporaneous relationship between stock returns and dp cannot be exploited for prediction

without knowledge of the future values of dp.

Demographic variables allow us to address both of these shortcomings. With the Census Bureau

projections for the MY ratio, we can construct recursive forecasts for dp that could be included in the

predictive regression for future stock returns. Some tentative findings from this exercise can be

summarized as follows. The model suggests that near-term returns are expected to be low by historical

standards until 2020 due to the projected fall of the MY ratio. After 2020, the MY ratio rises again,

with more middle-aged savers putting upward pressure on stock prices. Thus, after 2020, the long-

term stock returns are projected to increase again but settle at a level that is lower than the historical

average over the last 70 years.

While these projections reflect pure demographic information16 and are only suggestive about the

future long-term path of U.S. stock returns, they conform to a broader set of arguments put forward

in the literature. A consensus is now emerging that the changing demographic structure in the

developed economies has contributed to the recent decline in the equilibrium real interest rates

(Gagnon et al., 2016). Historically, periods of low real interest rates17 are associated with lower asset

returns in the next five years (Dimson et al., 2013). These lower expected returns then pose a direct

risk to institutional investors with long-term commitments.

15 This is based on ongoing work with Alex Maynard (University of Guelph) and Elena Pesavento (Emory University). 16 We acknowledge that the middle-young ratio could be just a convenient proxy for other slowly moving socioeconomic and political factors such as safety-net development and political polarization. Also, the demographic projections are based on assumptions about future immigration dynamics, which are influenced by policy decisions.

17 It should be noted that most of the previous episodes of low real interest rates were due to above-average inflation instead of low nominal interest rates, as in the years after the 2007–09 recession.

28

5. Concluding Remarks

This paper discussed and summarized some issues arising in the analysis of asset co-movements for

the purposes of asset allocation and portfolio diversification, as well as macroeconomic and macro-

prudential policy. One of the main findings that emerges from this review is that a judiciously

performed factor analysis appears to identify a common variation across domestic and international

asset classes at business and longer cycles. While the reported evidence is only tentative, a more

comprehensive empirical analysis with a larger cross-section of U.S. and international asset returns

would shed further light on the important sources of risks across asset classes, the integration of the

global markets, and their implications for diversification, re-allocation and policy analysis. Attaching a

risk factor interpretation to short- and medium-term co-movements appears to be more difficult due

to some statistical challenges in analyzing the data. Explicitly acknowledging the estimation and model

uncertainty as well as shifting the focus to more general measures of dependence would robustify the

decision-making process and reduce the risk of false positives/negatives in signal extraction and

performance evaluation.

One interesting topic that was omitted from the discussion is the increased importance of factor

(“smart beta”) investing. Studying more formally the dependence structure of these investment factors

and evaluating their performance using rigorous statistical criteria is an area of ongoing research. While

most of this research has focused on equities, constructing factors across divergent asset classes

enhances the ability of capturing multiple sources of systematic risk that are difficult to identify and

estimate statistically. Thus, a sufficient heterogeneity of spread factors across asset classes could

potentially span the underlying factor space (Roll, 2013) and contribute beneficially to risk

diversification and financial system stability.

29

References

Adrian, T., R. K. Crump, and E. Moench. 2015. “Regression-based estimation of dynamic asset pricing models.” Journal of Financial Economics 118: 211–44.

Anatolyev, S., and N. Gospodinov. 2010. “Modeling financial return dynamics via decomposition,” Journal of Business and Economic Statistics 28: 232–45.

Anatolyev, S., and N. Gospodinov. 2015. “Multivariate return decomposition: Theory and implications.” Atlanta Fed Working Paper 2015-7.

Ang, A., and J. Chen. 2002. “Asymmetric correlations of equity portfolios.” Journal of Financial Economics 63: 443–94.

Ang, A., and D. Kristensen. 2012. “Testing conditional factor models.” Journal of Financial Economics 106: 132–56.

Asness, C., A. Frazzini, N. J. Gormsen, and L. H. Pedersen. 2017. “Betting against correlation: Testing theories of the low-risk effect.” Working paper.

Assa, H., and N. Gospodinov. 2014. “Hedging and pricing in imperfect markets under non-convexity.” Atlanta Fed Working Paper 2014-13.

Baumeister, C., and L. Kilian. 2016. “Lower oil prices and the U.S. economy: Is this time different?” Brookings Papers on Economic Activity, Fall 2016: 287–336.

Baele, L., G. Bekaert, and K. Inghelbrecht. 2010. “The determinants of stock and bond return comovements.” Review of Financial Studies 23: 2374–428.

Bai, J., and S. Ng. 2004. “A PANIC attack on unit roots and cointegration.” Econometrica 72: 1127–177.

Bansal, R., and A. Yaron. 2004. “Risks for the long-run: A potential resolution of asset pricing puzzles.” Journal of Finance 59: 1481–1509.

Bassett, G. W., R. Koenker, and G. Kordas. 2004. “Pessimistic portfolio allocation and Choquet expected utility.” Journal of Financial Econometrics 2: 477–92.

Boyer, B. N., M. S. Gibson, and M. Loretan. 1999. “Pitfalls in tests for changes in correlations.” International Finance Discussion Papers No. 597, Board of Governors of the Federal Reserve System.

Brunnermeier, M., and L. H. Pedersen. 2009. “Market liquidity and funding liquidity.” Review of Financial Studies 22: 2201–238.

Campbell, J. Y., C. Pflueger, and L. M. Viceira. 2015. “Monetary policy drivers of bond and equity risks.” Harvard Business School Finance Working Paper No. 14-031.

Campbell, R. A. J., C. S. Forbes, K. G. Koedijk, and P. Kofman. 2008. “Increasing correlations or just fat tails?” Journal of Empirical Finance 15: 287–309.

Caporin, M., and M. McAleer. 2013. “Ten things you should know about the dynamic conditional correlation representation.” Econometrics 1: 115–26.

30

Cocharane, J. 2015. “Comments on ‘Robust Risk Premia’ by Michael Bauer and Jim Hamilton,” manuscript.

Cotter, J., S. Gabriel, and R. Roll. 2016. “Nowhere to run, nowhere to hide: Asset diversification in a flat world.” Working paper.

Cressie, N., and T. Read. 1984. “Multinomial goodness of fit tests.” Journal of the Royal Statistical Society B 46: 440–64.

Csiszár, I. 1972. “A class of measures of informativity of observation channels.” Periodica Mathematica Hungarica 2: 191–213.

Datta, D., B. K. Johannsen, H. Kwon, and R. J. Vigfusson. 2017. “Oil, equities, and the zero lower bound.” BIS Working Papers no. 617.

David, A., and P. Veronesi. 2016. “The economics of the comovement of stocks and bonds.” Chapter 15 in Handbook of Fixed-Income Securities, edited by P. Veronesi. Hoboken, NJ: John Wiley & Sons.

DeMiguel, V., L. Garlappi, and R. Uppal. 2009. “Optimal versus naïve diversification: How inefficient is the 1/N portfolio strategy?” Review of Financial Studies 22: 1915–53.

Dimson, E., P. Marsh, M. Staunton, and A. Garthwaite. 2013. Credit Suisse Global Investment Returns Yearbook 2013. Zurich: Credit Suisse.

Elliott, G. 1998. “On the robustness of cointegration methods when regressors almost have unit roots.” Econometrica 66: 149–158.

Embrechts, P., A. J. McNeil, and D. Straumann. 2002. “Correlation and dependence in risk management: properties and pitfalls,” Chapter 7 in Risk Management: Value at Risk and Beyond, edited by M. A. H. Dempster. Cambridge University Press.

Engle, R. 2002. “Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models.” Journal of Business and Economic Statistics 20: 339–50.

Favero, C. A., A. E. Gozluklu, and A. Tamoni. 2011. “Demographic trends, the dividend-price ratio and the predictability of long-run stock market returns.” Journal of Financial and Quantitative Analysis 46: 1493–520.

Favero, C. A., A. E. Gozluklu, and H. Yang. 2016. “Demographics and the behavior of interest rates.” IMF Economic Review 64: 732–76.

Fleckenstein, M., F. Longstaff, and H. Lustig. 2014. “The TIPS-Treasury bond puzzle.” Journal of Finance 69: 2151–197.

Forbes, K. J., and R. Rigobon. 2002. “No contagion, only interdependence: Measuring stock market

comovements.” Journal of Finance 57: 2223–261.

Gospodinov, N., and E. Maasoumi. 2017. “General aggregation of misspecified asset pricing models.” Working paper.

31

Gospodinov, N., P. Tkac, and B. Wei. 2016. “Are long-term inflation expectations declining? Not so fast, says Atlanta Fed.” Atlanta Fed macroblog, January 15, 2016.

Gospodinov, N., and B. Wei. 2015. “A note on extracting inflation expectations from market prices of TIPS and inflation derivatives.” Working paper.

Goyal, A., and I. Welch. 2008. “A comprehensive look at the empirical performance of equity premium prediction.” Review of Financial Studies 21: 1455–508.

Granger, C. W., E. Maasoumi, and J. C. Racine. 2004. “A dependence metric for possibly nonlinear processes.” Journal of Time Series Analysis 25: 649–69.

Hansen, L. P., and R. Jagannathan. 1991. “Implications of security market data for models of dynamic economies.” Journal of Political Economy 99: 225–62.

Herdin, M., N. Czink, H. Ozcelik, and E. Bonek. 2005. “Correlation matrix distance, a meaningful measure for evaluation of non-stationary MIMO channels.” IEEE VTC Spring 2005, 1: 136–40.

Jiang, L., K. Wu, and G. Zhou. 2016. “Asymmetry in stock comovements: An entropy measure,” Working paper.

Karolyi, A., and R. Stulz. 1996. “Why do markets move together? An investigation of US-Japan stock return comovement,” Journal of Finance 51: 951–86.

Longin, F., and B. Solnik. 2001. “Extreme correlation of international equity markets.” Journal of Finance 56: 649–76.

Patton, A. 2013. “Copula methods for forecasting multivariate time series.” Chapter 16 in Handbook of Economic Forecasting, edited by G. Elliott and A. Timmermann. Vol. 2, Part B. Elsevier.

Pukthuanthong, K., and R. Roll. 2009. “Global market integration: An alternative measure and its application.” Journal of Financial Economics 94: 214–32.

Roll, R. 2013. “Volatility, correlation, and diversification in a multi-factor world.” Journal of Portfolio Management 38: 11–18.

Song, D. 2017. “Bond market exposures to macroeconomic and monetary policy risks.” Review of

Financial Studies. Forthcoming.

Stutzer, M. 1995. “A Bayesian approach to diagnosis of asset pricing models.” Journal of Econometrics 68: 367–97.

Waggoner, D. F., and T. Zha. 1999. “Conditional forecasts in dynamic multivariate models.” The Review

of Economics and Statistics 81: 639–51.